High-Accuracy Numerical Solutions of Particle Motion in Static Magnetic Fields

TL;DR

The Parker-Sochacki method offers a highly accurate, stable, and efficient alternative to traditional Runge-Kutta methods for simulating particle motion in static magnetic fields, with significant improvements in energy conservation.

Contribution

This study demonstrates that the PS method outperforms RK and RKG methods in accuracy, speed, and stability across various magnetic field configurations for charged particle simulations.

Findings

PS method achieves 4 to 13 orders-of-magnitude better energy conservation.

PS method is faster than RK4 at matched energy error levels.

PS method maintains accuracy and stability for all tested particles and fields.

Abstract

The Parker-Sochacki (PS) method is investigated as an alternative to Runge-Kutta (RK) methods for solving the Lorentz equations of motion for a charged particle in a static magnetic field. Traditional methods, including fixed-time-step fourth-order RK, adaptive Dormand-Prince RK, and Gauss-Legendre Runge-Kutta (RKG), advance the solution by sampling derivative estimates at selected points to approximate the solution over a time increment. In contrast, the PS method uses a power series expansion in time that is specific to the system of equations, which is a fundamentally different approach. We assess the accuracy and long-term stability of the RK, RKG, and PS methods for three static magnetic fields: uniform, hyperbolic tangent, and dipole, with the RKG method included only for the dipole problem. The PS method results in a 4 to 13 orders-of-magnitude improvement in kinetic energy conservation over the RK methods. When the methods are compared at matched target kinetic energy error, the PS method was substantially faster than RK4, the method with the shortest runtime under identical fixed-time-step conditions. For the dipole field problem, the PS method had the lowest kinetic energy error and had runtimes 4 to 5 times shorter than RKG when using the same fixed time step for proton runs. The PS method was the only method in this study to maintain accuracy and stability for all problems for both protons and electrons; the RKG method failed on all electron runs in the dipole problem. We further show that, over sufficiently long integrations in inhomogeneous magnetic fields, the symplectic RKG may exhibit secular growth in energy error. Overall, these results indicate that the PS method provides a computationally efficient and highly accurate alternative to the symplectic RKG and standard RK methods.